Evolutionary Semigroups and Lyapunov Theorems in Banach Spaces

We present a spectral mapping theorem for continuous semigroups of operators on any Banach space E. The condition for the hyperbolicity of a semigroup on E is given in terms of the generator of an evolutionary semigroup acting in the space of E-valued functions. The evolutionary semigroup generated by the propagator of a nonautonomous differential equation in E is also studied. A "discrete" technique for investigating the evolutionary semigroup is developed and applied in describing the hyperbolicity (exponential dichotomy) of the nonautonomous equation.